# On an Equivalence Between Single-Server PIR with Side Information and Locally Recoverable Codes

## TL;DR

This paper reveals a fundamental equivalence between single-server PIR with side information and locally recoverable codes, enabling new bounds and insights into both areas.

## Contribution

It establishes a novel equivalence between PIR schemes with side information and locally recoverable codes, including cooperative variants, providing new bounds and theoretical insights.

## Key findings

- PIR schemes for single message retrieval are equivalent to classical LRCs.
- PIR schemes for multiple message retrieval are equivalent to cooperative LRCs.
- Derived new upper bounds on download rates for PIR-SI and cooperative LRCs.

## Abstract

Private Information Retrieval (PIR) problem has recently attracted a significant interest in the information-theory community. In this problem, a user wants to privately download one or more messages belonging to a database with copies stored on a single or multiple remote servers. In the single server scenario, the user must have prior side information, i.e., a subset of messages unknown to the server, to be able to privately retrieve the required messages in an efficient way. In the last decade, there has also been a significant interest in Locally Recoverable Codes (LRC), a class of storage codes in which each symbol can be recovered from a limited number of other symbols. More recently, there is an interest in 'cooperative' locally recoverable codes, i.e., codes in which multiple symbols can be recovered from a small set of other code symbols. In this paper, we establish a relationship between coding schemes for the single-server PIR problem and LRCs. In particular, we show the following results: (i) PIR schemes designed for retrieving a single message are equivalent to classical LRCs; and (ii) PIR schemes for retrieving multiple messages are equivalent to cooperative LRCs. These equivalence results allow us to recover upper bounds on the download rate for PIR-SI schemes, and to obtain a novel rate upper bound on cooperative LRCs. We show results for both linear and non-linear codes.